Noncommutative gauge theories on ℝ λ 3 $$ {\mathrm{\mathbb{R}}}_{\uplambda}^3 $$ : perturbatively finite models

Abstract

We show that natural noncommutative gauge theory models on R 3 λ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of R 3 λ and the components of the gauge invariant 1-form canonical connection. This latter object shows up naturally within the present noncommutative differential calculus. Restricting ourselves to positive actions with covariant coordinates as field variables, a suitable gauge-fixing leads to a family of matrix models with quartic interactions and kinetic operators with compact resolvent. Their perturbative behavior is then studied. We first compute the 2-point and 4-point functions at the one-loop order within a subfamily of these matrix models for which the interactions have a symmetric form. We find that the corresponding contributions are finite. We then extend this result to arbitrary order. We find that the amplitudes of the ribbon diagrams for the models of this subfamily are finite to all orders in perturbation. This result extends finally to any of the models of the whole family of matrix models obtained from the above gauge-fixing. The origin of this result is discussed. Finally, the existence of a particular model related to integrable hierarchies is indicated, for which the partition function is expressible as a product of ratios of determinants.

Closed star product on noncommutative ℝ 3 and scalar field dynamics

“…Some of these NCFT have been shown to be free of perturbative UV/IR mixing [54] and characterized by the occurrence of a natural UV cut-off, stemming from the group algebra structure underlying the R 3 λ algebra [55]. Gauge theories on R 3 λ have then been investigated [56,57]. The use of the canonical matrix basis introduced in [54] combined with suitable families of orthogonal polynomials together with a corollary of the spectral theorem [58] permits one to compute the propagator.…”

“…For reviews, see e.g [39][40][41]. Note that the Moyal plane can actually support causal structures stemming from Lorentzian spectral triples, as shown in [42]. Renormalizability of the model on R 4 θ is still unclear.…”

κ -Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight

“…This approach, which has been used in [35,36] is the one we mainly follow in this paper. This framework has also been used in recent studies on R 3 λ spaces [16,17] related to the convolution algebra of the compact SUð2Þ Lie group. Here, the relevant group is (related to) the affine group of the real line in the 2-dimensional case, i.e., a semi direct product of the two Abelian groups R, which extends in the…”

“…This latter corresponds, in the Moyal case, to the algebra for the Heisenberg group, which actually underlies the Weyl quantization, as it will be recalled below. For the space R 3 λ , it is the convolution algebra of SUð2Þ, which has been shown to play an essential role in originating the special properties of R 3 λ [14,16,17]. In the case of κ-Minkowski space-time, the relevant group algebra is the convolution algebra of the affine group as it will be shown below.…”

“…Doing this, one can take advantage of the machinery of the harmonic analysis on Lie groups and, in particular, measures involved in action functionals are provided by Haar measures. Note that such a viewpoint has also been intensively used in [16,17] for R 3 λ , the relevant group being SUð2Þ reflecting the suð2Þ noncommutativity of the quantum space and has provided the relationship between R 3 λ and the convolution algebra of SUð2Þ, the determination of the natural measure in the action functionals and by the way clarified the origin of the matrix basis used in [14]. In the case of κ-Minkowski space, we note that such a natural construction has already been used in [35,36] to derive a star product for a 2-dimensional κ-Minkowski space and to characterize a related multiplier algebra [35].…”